In the modern power grid, distributed generators are widely connected to the grid via voltage source converters (VSCs). Analyzing the transient stability of VSCs under large disturbances is useful for maintaining the security of the grid. However, this topic is very little studied. In this paper, the transient stability behavior of the droop-controlled VSC is theoretically explained. In particular, it is shown that transient instability can occur to the droop-controlled VSC when its current is saturated under large disturbances. In addition, the dynamics of the VSC under voltage sags that could incur instability problem is elaborately studied to consider special non-fault disturbances. To deal with this instability problem, a stability enhanced P-f droop control is proposed. Simulations and hardware-in-the-loop experiments verify the validity of the transient stability analysis and the effectiveness of the proposed control scheme.